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2.20.2012

Next year is my last year of having an instructional coach so I really want to get all of my big ideas out there on the table by next year so that I have some kind of help and guidance when things go wrong.

I am always thinking about SBG and how I can make it work. I re-read my last post on SBG, which was almost a year ago. I think that might now be possible. My math teacher counterpart is leaving after this year and I'm thinking the administration might consider hiring the current student teacher. If so, it will be their first year and so far they seem really...'moldable'. And I think any first year teacher would love to have another teacher give them resources, ideas, and collaboration. If this is true, we might be on the same page with SBG and so on!

Anyway, my brainstorm is this. My number one struggle with SBG has been not assessing everything I teach. In my mind, I feel like if I'm teaching it, why wouldn't I assess it? Now I'm thinking that I just need to cluster. Concepts build on top of each other toward an ultimate goal. If I only assess that ultimate goal, then it requires knowing the concepts that build up to that goal without assessing every little thing individually. I would call this ultimate goal a performance event and use a rubric idea to grade it. A perfect score would be getting every skill correct. Otherwise, each skill would move you one step higher on the rubric.

Here's my example based on a mini-unit of solving systems of linear inequalities. The ultimate goal of the unit would be to graph a system of inequalities from a word problem.

Skill 1: Write a system of inequalities from a word problem.
Skill 2: Solve each inequality for slope-intercept form.
Skill 3: Graph both lines of the system (dotted or solid).
Skill 4: Plug in a test point and shade.
Skill 5. Choose an ordered pair from the solution set.

To earn a 5 for the ultimate goal, they would have to correctly complete skills 1-5. If they could only do 1 and 2, their score would be a 2. By comparing the rubric and the students grade, it should still be easy to remediate.

Here's where my idea breaks down. What if they complete every skill but they screwed up on skill 2? If they complete the subsequent skills based on their answer for skill 2, they may have done the process correctly but still ended up with a wrong answer. Would they then get a 2 because that is the level where they screwed up? Or would they get something like a 4.5 because they did the process correctly? Translating to the gradebook, if each skills was worth 20%, would the student get a 40% (2) or a 90%(4.5)? That's a huge difference.

Another option would be to count each skill as one point (20%) so that they get credit for every skill they performed correctly but then their score would tell me nothing about where they messed up. :(

Other than that, I am liking this idea. The individual skills will be assessed throughout the unit through class work and homework quizzes but the unit test would now be replaced with this performance event idea.

I also want to try math portfolios next year. I'm thinking (hoping) that this idea would naturally tie into the portfolio. After completing the performance event, the student would then write a structured reflection/explanation of the process. Throughout the year, these performance events and reflections would be collected as a demonstration of mastery.

One last thing we are working on are End-of-Course tests. This is not a state mandate but a district suggestion. We want it. We have currently finished our Algebra I EOC minus a few edits. We will give this four times a year to show growth over time. This will also help students be prepared for long, standardized tests as opposed to the short performance event quizzes covering only one concept.

Please tell me where you see potential downfalls and advise me on how to grade!

2.12.2012

I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

I really really liked this chapter so I'm pretty much typing the whole thing. But oh well!

For the purposes of this chapter, I am defining discourse as the genuine sharing of ideas among participants in a mathematics lesson, including both talking and active listening.

What Discourse Looks Like

Traditional. Prompts from the teacher generally lead to a preplanned answer. I do this.

Probing. Stem from the teacher's desire to hear about students' thinking, rather than from a need to move students along a planned route. I do this too.

Discourse-Rich. "Students must...learn to question and probe one another's thinking in to clarify underdeveloped ideas. Not this.

When the classroom climate fosters genuine student discourse, students react to their classmates; ideas, asking questions and checking for understanding. Conversations continue without the need for teacher participation or supervision.

Creating Discourse-Friendly Classrooms

Arrange furniture so that students can easily turn to see each other. They must be able to speak and listen to classmates. How can I do this with 24 desks?

Encourage students to direct questions and explanations to the class, rather than to the teacher. I usually redirect the question back to the class but have not actually told to ask the class instead of me.

When recording ideas on chalkboard or chart paper, use the students' words as much as possible. This is a matter of respecting their ideas.

Try not to repeat or paraphrase everything students say. Paraphrasing can give the impression that the student is being corrected and may indicate to others that they don't need to listen unless the teacher speaks.

Remind students that conversation is a two-way operation requiring both talking and listening.

Stand in a variety of spots. As students turn to look at you, their views of the classroom and their positions relative to classmates will shift.

Give students time to think. Wait time or brief writing moments help students to solidify ideas and formulate good questions.

Arrange lessons so that students have a product to share as they explain their thinking. They might illustrate ideas on chart paper or overhead transparencies or demonstrate by using manipulative materials.

Teachers must understand mathematics at a deep level in order to follow young people's unconventional reasoning. This is exactly what I am afraid of!

Here are several ways to let student ideas take the lead in class:

Involve students in engaging and challenging problems.

Ask open questions to stimulate student thinking. (Examples: "What does this make you wonder about?" "Are there patterns?" "Is this logical?" "Can we estimate a solution?")

Listen carefully to student responses.

Train students to listen to their classmates; observations by asking questions that engage.

Honor diverse ideas, methods, and example from varied sources.

Honor ideas even if they're incorrect. Do not quickly agree or disagree. Students will come to realize that you are giving them time to think and to justify. Often, as students explain erroneous thinking, they uncover their own errors or classmates step in to clarify or correct them.

Encourage mathematical arguments between students.

Remember, confusion is okay. Some of the best learning happens when we sort out what it is that has pushed us a bit out of balance. Be sure students know you are deliberately letting them be confused and that this is based upon your knowledge of how people learn, as this tactic may not match what previous teachers have done. And they will think you are not doing your job!

Take time to let students share different problem-solving methods. Even when a correct solution has been shown, ask if there are other ways to do the problem. This helps to deepen understanding and makes students more willing to work with their own strategies, rather than thinking there is only one correct method.

Capture teachable moments. Tangents are good.

Decide how much leadership your students need. Let students' ideas lead but this doesn't mean that the class just moves without teacher direction.

Tips for Facilitating Classroom Discussion

Focus students attention on a problem, puzzle, figure, process, question, or set of numbers. Stimulate discussion by asking the following types of questions:

What do you notice?

Do you see any patterns?

What is similar?

What is different?

How do you think this works?

Why does this work/look this way/give this result?

What questions do you have?

What can we do with this information?

What do you want to know?

Rather than rephrasing their responses, ask, "How do you want to say this?"

When observations or questions are brought up by one student, ask, "What do the rest of you think about this idea? Does it make sense?" Encourage them to consider other examples that would show that the observation is or is not always true.

Motivate students to search for patterns, delve deeper, and generalize.

If students are making mistakes or doing something awkwardly, ask them "Is there an easier or more efficient way?" or "In what other ways could this be done?" rather than telling them how to do it.

If students have difficulty thinking about a concept, suggest examples to consider or play devil's advocate. Ask "What if?" questions. I love playing devil's advocate!

Counter questions with questions instead of explanations. (Students hate this!) People tend to blank out when one person asks a question and the teacher immediately gives an explanation.

Even when a solution is successful, take time to ask whether anyone did the problem a different way or discarded an idea. Help students to build confidence in their own ideas , knowledge, and insights by showing that problems can be solved in a variety of ways.

Discourse and Problem Solving

As students explain their thinking, others can see connections and the usefulness of different methods.

Summarizing and labeling the strategies make them memorable, as does naming them in honor of the students who came up with them (e.g., Mary's Method, Theo's Theory, Pedro's Plan).

Discourse and Vocabulary

During lessons in which students first encounter a new concept, teachers should encourage them to describe ideas in their own words before introducing the specialized terms.

Summary

As we change our teaching, it is important to realize that we expect students to change with us; they also have new responsibilities.

Students may well be uncomfortable shifting from passive observers to active learners. How can we help them learn these new skills?

I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

The following aspects of mathematical language are particularly confusing to students:

Technical symbols such as ∑, ≤ , or Δ. These signs, also known as logograms, stand for whole words but have no sound-symbol relationship for students to decode.

Technical vocabulary- words such as rhombus, hypotenuse, and integer, which are rarely used in everyday conversation.

The assignment of special definitions to familiar words such as similar and prime.

Subtle morphology (one hundred, hundreds-place, hundredths) and the use of "little words" (prepositions, pronouns, articles, and conjunctions) in a technical syntax so precise that meaning is often obscured rather than clarified.

Scenario #1: Measuring Cups Activity

*A teacher models drawing a one cup and one-fourth cup four times to quadruple a recipe. When asked how many cups are needed, students responded 'eight' because there were literally eight cups in the drawing. They counted the one-fourth cup the same as the one cup.

In what other ways might the students have attempted to model the problem if the teacher hadn't offered the initial suggestion? Let students try first, then offer suggestions?

How might using actual three-dimensional models (i.e., a set of nested measuring cups) before using two-dimensional representations alter the transfer of learning?Always teach from concrete to abstract?

Scenario #2: A Round Pizza in a Square Whole

*When solving a problem dealing with rectangular pieces of pizza, William drew the problem with a circle pizza. To him, fractional parts had to be round.

How can the teacher help [William] move beyond a single conceptual image and experiment with new metaphors and visual models?

When are models useful, and when do they get in the way of new learning?

Scenario #3: An Uphill Struggle (Slow on a Slippery Slope)

*When discussing slope from points on a graph and connecting points to create intervals, a student took the graph literally, thinking the the intervals were literally hills and valleys.

For how long, and to what depth, should a teacher continue to probe in order to get to the logical and intuitive root of confusion?

Scenario #4: Breaking Even

*Students looking for the break even interval on a graph of monthly profits picked the flat interval because they were thinking break even literally meant to find the interval that is even.

Note that the phrase breaking even is not a formal mathematical term but rather a conversational idiom with mathematical implications. Pimm (1987) calls these types of phrases locutions- "certain whole expressions whose meanings cannot necessarily be understood merely by knowing the means of the individual words, that is, the expressions function as semantic units on their own" (p.88).

How can teachers become more aware of the mathematical locutions embedded in their classroom conversations? Pre-teach all mathematical terms, especially if they are also used in a nonmathematical context? Think literally.

Would this particular mathematical conversation have occurred if the students didn't have a graphic "prop"?

How did the visual representation act as a catalyst for student discourse?

How might conversations about a graphic display encourage students to put their mathematical perceptions into writing?

Now that a conversation had begun on the topic of exponential decay, how might the teacher build on the richly descriptive terms that the students created to press for more precise mathematical terminology?

Scenario #7 Where's the Fourth Fourth?

*Benjamin folded a paper strip into fourths and wrote on each of the folds 1/4, 2/4, and 3/4. When asked what his strip created, he said thirds. He was judging each strip based on the last numerator instead of the denominator. When asked about his strip that was folded in half, he wanted to say halves but according to his system, it would be one whole, based on the one in the numerator of one-half.

What distinction was Benjamin making between the terms thirds, three folds, and dividing in to three parts?

How might students conceptualize fourths differently, depending on whether they are asked to label the strip's segments (1/4, 2/4, 3/4, 4/4) or its folds (1/4, 2/4, 3/4)?

What consideration might the teacher need to give to precision of language when providing directions for this task?

What role might peer-to-peer discourse have played in helping Benjamin test his conjecture?

What questions might a teacher ask to check more deeply for understanding if, at first glance, a student's thoughtfully done work is apparently correct? Ask them how they would explain it to their grandma or ask them to create a new example?

Scenario #8: Three Three/Ten Combos

*Benjamin and his class were instructed to divide a square pan of brownies into 30 equal squares. Students drew 5 rows of 6, 3 rows of 5, and 2 rows of 15. Later they were given a sheet with a giant square divided into 10 vertical strips. When asked to model three-tenths, Benjamin drew three horizontal lines across the vertical strips instead of shading in 3 of the 10 strips.

From what assumptions might Benjamin have been working when he approached this new concept?

Where was there a language-based component in his confusion?

What do the phrases "three-tenths" "three and tenths", and "thirds and tenths" mean mathematically? What do they seem to mean to Benjamin?

In a roomful of 20 students, how difficult is it to hear the subtleties of different word forms?

What might Benjamin's drawing imply about what he heard?

How could the teacher use Benjamin's drawing to encourage him to express his personal understanding in words?

How might making additional sketches have helped Benjamin communicate what he head and understood?

Scenario #9: Holes in Her Logic
*Sarah was dividing donated food into boxes for 24 families. She drew 24 squares and drew dots in each box until she ran out so that they would be evenly distributed. For 12 pounds of cheddar cheese, Sarah divided 12 by 24 in the calculator and got .5, so she then drew 5 dots in each box instead of half of a dot.

What windows to Sarah's thinking do her cocoa, milk, and cheese drawings provide?

How did Sarah seem to understand division in general? What about dividing in situations when the quotient is less than one?

What did Sarah believe about the decimal point?

What was Sarah hearing, and how did this influence the manner in which she conceptualized decimal fractions?

What is appropriate calculator use for students beginning to work with fractions and decimals?

Would language-based cues, such as the teacher's asking how to divide a 12-pound wheel or chunk of cheese among 24 families, suggest useful visual models? Or might metaphors create further confusion? If I was Sarah and the words wheel or chunk were mentioned, I would have drawn wheels or chunks instead of dots but I still would have drawn 5 wheels or chunks in each box.

What might the teacher do ti reinforce the use of proper terminology when students are working with decimal fractions?

I don't feel like typing Scenarios #5 and #10.

When teachers are asked to reflect on how student drawings can inform their practice, three themes emerge. Teachers feel that the drawings:

Make the students more aware that they're "speaking mathematics" in class,

Show a need for greater precision in the students' use of mathematical language, and

Suggest areas in which directions and explanations should be more clearly phrased.

Suggestions from Teachers

Combine verbal with visual.

Monitor your language for words with double meanings.

Assume positive intent to understand even from silly questions or offhand remarks.

Ask students to move from drawing detailed pictures to simpler shapes, another step toward abstraction.

Consider the sequence of representations students select. Does it promote depth of mathematical thinking?

Actively point out connections to other representations so that students become fluent in translation from one representation to another.

Suggested Practices

Articulate and enunciate!

Keep writing utensils available if you expect students to draw.

Extra time: putting ideas on paper takes more time than talkingg.

For open-ended questions, suggest drawing a diagram first in order to have something concrete to write about.

Make copies of important textbook pages so that students can highlight, draw on, underline, and write notes in the margin to better interact with the text.

Have students use graphic organizers that incorporate both words and pictures.

Designate a section of notes/worksheets for drawing, showing that you value and expect to see visual thinking.

Drawing is a device to capture the language of mathematics in order to make it visible to themselves.

Drawing slows students down and allows them to self-correct their thoughts while their hands are sketching; it also helps them to keep track of and record their solutions.

2.07.2012

Because my school is a SIG (School Improvement Grant) school we are periodically observed by the Illinois State Board of Ed. A couple weeks ago, they came in to observe the math and English departments. I was observed on Wednesday and the English teachers were scheduled to be observed on Thursday. Due to a freak break in our water main, the school had no water and was shut down on Thursday. So ISBE's only impression of our teachers was based on my classroom...and I rocked it. Apparently they bragged all over me and were impressed at how rigorous my lesson was considering I am only a third year teacher. Plus, they missed the first half of my class! They were in a meeting and someone interrupted them to tell them they were supposed to be in my class. They said they shouldn't tell teachers when to expect them and then not show up! lol

Anyway, they had asked me ahead of time to prepare lesson objectives, how my lesson connected to the Common Core, sample student work, and any formative or summative assessments that would follow the lesson. I will say that I went overboard in creating a lesson plan with a fancy template and I decorated the folder I put everything in. I also had a student greeter who welcomed them in the room and escorted them to their seats. I think these were the things they were most impressed by.

As far as my actual lesson, I feel confident that it was not posed or over the top but included things I normally do. I didn't prep my kids other than to tell who would be there and I felt that the classroom environment was essentially unchanged after they took their seats.

So let me preface: we had just learned the square root method to solving quadratic functions. The day before I had quite a few students out so we did some board work and review. I wanted to briefly introduce the formula to get the following day's lesson off to a good start. I decided to try a 3D puzzle, which is something we had tried in our grad class. What I did was create an 'empty' quadratic formula and overlay that with a 5 x 5 grid (I have 5 students) in Powerpoint.

I cut them apart into 5 strips of 5. Each student had a strip and two crayons. No students could have the same two shades of crayons. They had to use both colors in each square on their strip and no space could be left white. Then we cut them all apart, mixed them up, and they had to put them together.

I thought it would be easy but it did take longer than expected and it required them to work together. I ended class by showing them this classic video- if you watch it, you have to watch the whole thing.

So the next day, we started class as usual with a bell ringer. I asked them to write the standard form of a quadratic and the formula for finding the AOS of an unfactorable quadratic. I asked the students to explain what we did the say before with the puzzle and then we all watched the video again. Now, everyone has seen the formula. We started our notes by writing down the quadratic formula and the AOS formula and comparing. They notice that both formulas have -b/2a. Here's where I transition into the discriminant. I tell them we are going to ignore the part of the formula we are familiar with and work with the b^2 - 4ac. I send them to the board and they do three example problems, just finding the discriminant. Then it's back to their seats. I give them a baggy with six different equations.

I tell them to find the discriminant of each (which they have already shown me they know how to do at the board) and then I ask them to sort. I love sorting! I don't give them any information. Some students sorted them into 2 piles of three, based on the equations set equal to y or 0. Some looked at where the 3's and 4's were and sorted by their positions. Then I told them they had to have three piles. They re-sorted. I asked each of them to describe how they sorted in one sentence. They shared and we decided who had the best idea. They agreed that one pile had negative answers, one was positive, and the others equaled zero. That led us into a discussion about real and imaginary roots and visualizing what roots mean and whatnot.

Next I said that we would put the old part of the formula and this new part together and practice using the entire formula. We consulted our sorting cards to tell us how many roots the first equation should have and students set off to solving. This was the first time so all the problems worked out to be nice whole numbers and no negative discriminants. I ran out of time before finishing all of the examples, but like a good girl, I stopped early to do the exit slip, which was asking them to answer the lesson's essential question: "How does the quadratic formula help you find the roots of a quadratic equation?" I thought it tied in nicely with the unit's essential question: "How can we find the roots of any type of quadratic equation?", both of which I made up myself. My observers came in late, after the video and board work but right during the sorting. I love sorting! Luckily, I got to talk to them afterwards and explain my cool 3D puzzle and show them the Crank Dat Quadratic Formula video. One of them is a former math teacher and was familiar with the Pop Goes the Weasel tune but liked the new video. She even said, "Isn't that that Superman song?" which earned her brownie points with me. They asked me if I created the lesson template myself, which I did, but was inspired by one I found from Microsoft Word. They thought that it looked very much like backwards design which made me happy inside because that's where I'm trying to go. Since then we have solved quadratics with like terms on both sides, with discriminants that need to be simplified, finding exact and approximate answers, quadratic applications and word problems. I feel that they have handled it all really well and that this was a great start. I'm no Dan Meyer but I'm proud of my lesson and it's results and I wanted to share that all with you! While I'm sharing, here are all my resources that I mentioned above: Lesson Plan TemplateUnit Plan Template (I did not create) Quadratic Formula 3D PuzzleDiscriminant Sort (I love sorting!) The Quadratic Formula PPTThe Quadratic Formula NotesBell Ringer and Exit Slip
Thanks for reading my extra long post and for cheering me on.